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Theorem ineccnvmo 38358
Description: Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 2-Sep-2021.)
Assertion
Ref Expression
ineccnvmo (∀𝑦𝐵𝑧𝐵 (𝑦 = 𝑧 ∨ ([𝑦]𝐹 ∩ [𝑧]𝐹) = ∅) ↔ ∀𝑥∃*𝑦𝐵 𝑥𝐹𝑦)
Distinct variable groups:   𝑥,𝐵,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧

Proof of Theorem ineccnvmo
StepHypRef Expression
1 relcnv 6122 . . 3 Rel 𝐹
2 id 22 . . . 4 (𝑦 = 𝑧𝑦 = 𝑧)
32inecmo 38356 . . 3 (Rel 𝐹 → (∀𝑦𝐵𝑧𝐵 (𝑦 = 𝑧 ∨ ([𝑦]𝐹 ∩ [𝑧]𝐹) = ∅) ↔ ∀𝑥∃*𝑦𝐵 𝑦𝐹𝑥))
41, 3ax-mp 5 . 2 (∀𝑦𝐵𝑧𝐵 (𝑦 = 𝑧 ∨ ([𝑦]𝐹 ∩ [𝑧]𝐹) = ∅) ↔ ∀𝑥∃*𝑦𝐵 𝑦𝐹𝑥)
5 brcnvg 5890 . . . . 5 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦𝐹𝑥𝑥𝐹𝑦))
65el2v 3487 . . . 4 (𝑦𝐹𝑥𝑥𝐹𝑦)
76rmobii 3388 . . 3 (∃*𝑦𝐵 𝑦𝐹𝑥 ↔ ∃*𝑦𝐵 𝑥𝐹𝑦)
87albii 1819 . 2 (∀𝑥∃*𝑦𝐵 𝑦𝐹𝑥 ↔ ∀𝑥∃*𝑦𝐵 𝑥𝐹𝑦)
94, 8bitri 275 1 (∀𝑦𝐵𝑧𝐵 (𝑦 = 𝑧 ∨ ([𝑦]𝐹 ∩ [𝑧]𝐹) = ∅) ↔ ∀𝑥∃*𝑦𝐵 𝑥𝐹𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wo 848  wal 1538   = wceq 1540  wral 3061  ∃*wrmo 3379  Vcvv 3480  cin 3950  c0 4333   class class class wbr 5143  ccnv 5684  Rel wrel 5690  [cec 8743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rmo 3380  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747
This theorem is referenced by:  ineccnvmo2  38361
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